Question

To determine whether the ordered triple is a solution to the system.

$\left\{\begin{array}{l}2x-6y+z=3\\ 3x-4y-3z=2\\ 2x+3y-2z=3\end{array}\right.$

(a) $(3,1,3)$

(b)$(4,3,7)$

Short answer

Part (a) The given ordered triple $(3,1,3)$ is not a solution to the system of linear equations.

Part (b)The given ordered triple localid="1658928875389" $(4,3,7)$ is not a solution to the system of linear equations.

Step-by-step solution

Part (a) Step 1. Given information

We have been given equation $\left\{\begin{array}{l}2x-6y+z=3\\ 3x-4y-3z=2\\ 2x+3y-2z=3\end{array}\right.$

and coordinate$(3,1,3)$

Part (a) Step 2. Testing

Consider the given equation:

$\left\{\begin{array}{l}2x-6y+z=3\\ 3x-4y-3z=2\\ 2x+3y-2z=3\end{array}\right.$

We are given:

$\left\{\begin{array}{l}x=3\\ y=1\\ z=3\end{array}\right.$

Substituting this coordinates in given equation and checking if true.

First substitute into the equation $2x-6y+z=3$

$\begin{array}{r}2x-6y+z=3\\ 2\left(3\right)-6\left(1\right)+3=3\\ 6-6+3=3\\ 3=3\end{array}$

This is true.

Also substitute the values into the Second equation.

$\begin{array}{r}3x-4y-3z=2\\ 3\left(3\right)-4\left(1\right)-3\left(3\right)=2\\ 9-4-9=2\\ -4=2\end{array}$

This is not true.

The given ordered triple is not a solution of the system of linear equations.

Part (b) Step1. Given information

Given equation:

$\left\{\begin{array}{l}2x-6y+z=3\\ 3x-4y-3z=2\\ 2x+3y-2z=3\end{array}\right.$

and coordinate:$(4,3,7)$

Part (b) Step 2. Testing

We have:

$\left\{\begin{array}{l}x=4\\ y=3\\ z=7\end{array}\right.$

Substitute these values in equation one.

$\begin{array}{r}2x-6y+z=3\\ 2\left(4\right)-6\left(3\right)+7=3\\ 8-18+7=3\\ -3=3\end{array}$

Not true.

The given ordered triple is not a solution to the system of linear equations.